where \(U_{i,j}\)  are the elements of the \(2~ x ~2\)  unitary matrix \(U\) .
We now replace \(O_{ l,m}\)  in Eq. 7 with \(O^{+}_{l,m}\) , and \(O_{- l,m}\)  in Eq. 19 with \(O^{-}_{l,m}\) , and rewrite the two perturbation series in a standard procedure [1]}, [2]}, [3]} as
\(O^{\prime \pm }_{l,m}=O^{\pm }_{l,m}+\sum _{i=1}\sum _{n \ne \pm l,k}a^{\pm (i)}_{(l,m)(n,k)}O_{n,k},\) 
